Axiom:Axiom of Foundation

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Axiom

For all non-empty sets, there is an element of the set that shares no element with the set.

That is:

$\forall S: \paren {\paren {\exists x: x \in S} \implies \exists y \in S: \forall z \in S: \neg \paren {z \in y} }$

The antecedent states that $S$ is not empty.


Also defined as

It can also be stated as:

For every non-empty set $S$, there exists an element $x \in S$ such that $x$ and $S$ are disjoint.
A set contains no infinitely descending (membership) sequence.
A set contains a (membership) minimal element.
The membership relation is a foundational relation on any set.


Also known as

The axiom of foundation is also known as the axiom of regularity.


Sources